How do I check to see if my object unwrapped properly after export?
I would suggest making sure you have the up and forward re-assigned in the export screen, For Example in FBX export at the lower left hand side:



I would suggest creating an 'Operator Preset' once you have the right export result.
Please don't take this screen shot as the exact settings you need. You would for example have 'Selected Objects' probably ticked for what you are trying to do.

There are a range of other tools and programs to re-open the file.
You could try importing directly into Unreal Engine or Unity. Perhaps even GODOT, (Godot tutorial here)

There is also 3D model review software like:
Autodesk FBX Review

I hope this helps.